Advanced Visual Quantum Mechanics

Interactive engagements for use in upper-level undergraduate quantum mechanics courses (Revised 8/9/01).

(Click here to download everything on this page.)


All of the materials on this page are currently under development but we would love to have you test them in your quantum mechanics courses.  Feel free to download any of the material and use it as you see fit.  If you do use something, please contact us at and tell us what you did and how it went.  We also have a survey and a testing instrument (Robinett’s QMVI from Penn State) that we would like you to give your students (and an informed consent document so we can use the data they provide).  Also feel free to contact us with any questions.


Quick Start:

For a quick view of an AVQM program, take a look at Wave Function Explorer(currently only runs well in Internet Explorer for Windows).  In it’s sketcher mode, this program allows you to sketch eigenfunctions of the time independent Schrödinger equation in piece-wise constant potentials.  In it’s explorer mode, it animates the stationary state wave functions in the same potentials.  The Shape and Behavior of the Wave Function materials described below are based on this program.


The Materials:

Currently all of the materials (except Wave Function Explorer and Wave Packet Explorer) must be downloaded to your local computer to be used.  The written materials are Word documents, Wave Function Explorer is a java applet, and Classical Probability Explorer is a self-installing Visual Basic application (there are also some QuickTime movies for use with this program).  We hope to have everything converted to html, pdf, and java later in the semester.


Preliminary materials

These materials are designed to help introduce some preliminary concepts near the beginning of a course.  There are currently three main topics:  Energy Diagrams, Classical Probability, and Introduction to the Uncertainty Principle.


Energy Diagrams:  In all of our previous experiences teaching VQM, we found that students needed extensive help understanding the concepts behind energy diagrams.  These more advanced students probably do not need nearly as much help in this area.  There are two forms of these materials.  The long form has two parts: Energy Diagrams I and Energy Diagrams II.  These are minor rewrites of the materials we use with our modern physics classes.  The short form is a greatly condensed version of this material and has just one part: Energy Diagrams Short.  Both sets of documents use the Shockwave program Energy Diagram Explorer.


Classical Probability:  The two documents: Classical Probability I and Classical Probability II use video and discussion of classical motions to help the students learn about probabilities in a familiar context.  These interactive engagements take the students from what they know about motion in terms of velocity and time spent in a region to the “new” concepts of probability and probability density.  Then the relationship between probability density and the potential energy (for classical motion) is derived.  Finally, concepts are solidified through interactive computer experiments using videos of some simple classical situations.  These computer experiments use the Visual Basic program: Classical Probability Explorer and some QuickTime movies.  There is also a short document about How to Use Classical Probability Explorer.


Introduction to the Uncertainty Principle:  This is a short section adapted from our modern physics materials that gives a basic introduction to the uncertainty principle.  There is one document: The Uncertainty Principle Part I, and one Shockwave program: Wave Packet Explorer.  More in-depth materials are planned for later development.



The Shape and Behavior of the Wave Function

These are the materials that we would most like to have tested.


The Shape and Behavior of the Wave Function is a series of seven interactive engagements (in-gagements) and their supporting materials designed to help the instructor guide the student through learning about most of the ideas of one-dimensional quantum physics.  It strives to build the student’s basic knowledge, visual intuition, and analytical skills by following a modified learning-cycle format with lots of hands-on activities and exercises.  It is designed to be used as a supplement to any standard undergraduate quantum physics text.  A list of its parts, with descriptions, follows:


Wave Function Explorer (WFE) (download, run online) WFE is a powerful java applet designed to be used with the AVQM: Shape of the Wave Function materials.  In it’s sketcher mode, this program allows you to sketch eigenfunctions of the time independent Schrödinger equation in piece-wise constant potentials.  In it’s explorer mode, it animates the stationary state wave functions in the same potentials.


Introduction:  A brief introductory document containing goals, prerequisites, introductions to main ideas and math, and an overview of the seven interactive engagements (in-gagements).


The seven in-gagements (download) : The first six in-gagements deal with quantum physics in piecewise constant potentials.  These in-gagements step the student through all of the main ideas of one-dimensional quantum physics in the context of these “easier” physical situations.  Each of these in-gagements is divided into three parts: Getting Started, Exploring, and Applying and Expanding.  The Getting Started section contains a brief introduction to the main ideas and mathematics of the in-gagement along with simple exercises and conceptual questions designed to get the student thinking along the right lines.  The Exploring section contains a series of computer-based activities designed to guide the student through developing concepts and visual intuition.  The Applying and Expanding section contains a summery of the in-gagement and a series of more traditional exercises and questions designed to more fully develop the student’s understanding.  (One way to use these three parts would be to give the Getting Started section as homework, do the Exploring section as in-class group work, then assign the Applying and Expanding section as more homework (along with the next Getting Started section).  The final in-gagement helps the student to transfer all of the ideas they have learned about piecewise constant potentials to arbitrary one-dimensional problems.


The in-gagement titles are listed below along with a list of the main ideas they introduce:


The Free Particle(Getting Started, Exploring, Applying and Expanding): Separation of variable, stationary states, basic solutions, standing waves, traveling waves.


The Step Function(Getting Started, Exploring, Applying and Expanding): Multiple regions, boundary conditions, free parameters, superpositions, scattering, reflection and transmission coefficients.


General Scattering and Resonance(Getting Started, Exploring, Applying and Expanding): Multiple boundaries, general scattering, shape resonance.


Tunneling(Getting Started, Exploring, Applying and Expanding): Basic solutions for E<V, tunneling.


Half Scattering(Getting Started, Exploring, Applying and Expanding): Asymptotic conditions, ‘infinite’ potentials, resonance (pseudo bound states).


Bound States(Getting Started, Exploring, Applying and Expanding): Eigenvalues, discrete spectra.


General Potentials: Quantum physics in arbitrary one-dimensional situations, sketching wave functions, approximation techniques. (uses an Excel worksheet)


I hope you enjoy the materials and find them useful – let me know if you have any questions.


Wally Axmann